## Tuesday, February 7, 2023

This post is a part of the series on audio amplifier feedback. The contents of the series can be found here.

A bootstrapped collector load is a pair of resistors connected in series, with their common point actively driven, often by a unity gain buffer via a large capacitor:

Bootstrapped loads have been popular in audio amplifiers for more than 50 years, although these days they are largely replaced by constant current sources.

The work of a bootstrapped load is easily understood. Because of the unity gain buffer, the right side of the capacitor $C$ (see the schematic above) sees the same potential as the bottom end of $R_2$. As long as the voltage across the capacitor is relatively stable, the voltage across $R_2$ is relatively stable, and so is the transistor's collector current flowing through $R_2$. In particular, the collector current changes only a little with changes in the collector potential (e.g. an audio signal), as if the collector load resistance is large.

Quantitatively, it is obvious that at DC, $R_1+R_2$ is the only load that sets the quiescent collector voltage. To find out AC response, one needs to e.g. write down node equations, which reveal that a bootstrapped load presents a frequency dependent impedance of the form $$Z = (R_1 + R_2) \times (1+s T_z)$$

where $T_z = (R_1 || R_2) C$.

The impedance increases with frequency at 20dB/decade, as would the impedance of an inductor. Effectively, a bootstrapped load is a synthesized inductor with the series resistance $R_1+R_2$ and the inductance $L = R_1 R_2 C$. For example, with R1=R2=1kOhm and C=10uF, the equivalent inductance L=10H.

In a real circuit, this synthesized inductor is not the only collector load - connected in parallel to it are the output impedance of the transistor and the input impedance of the buffer. If this input and output impedances are lumped into Ri:

then the combined collector load has the form $$Z = {R_i || (R_1 + R_2)} \times {{(1+s T_z)} \over {(1+s T_p)}}$$ where $T_z = (R_1 || R_2) C$ as above and $T_p = {{ (R_1 + R_2) \over (R_i +R_1 + R_2) } C}$.

I will look more into the behavior and practical uses of a bootstrapped load in separate posts.